I'm stuck on this question:
Q: The outer conductor (radius b, length l) of a long cylindrical capacitor is earthed. The inner conductor (radius a, length l) is hollow, insulated and uncharged. A sphere of radius R is charged to a potential V far from any other bodies and is inserted inside the inner conductor of the cylindrical capacitor without touching it. (The length l is very much greater than R so end-effects may be ignored). Draw diagrams showing the distributions of the induced charges and the E-field lines in two perpendicular planes through the centre of the sphere, one parallel and one perpendicular to the axis of the cylinder. Show that the electri field strength at a radius a<r<b is E=2RV/rl and hence find the potential of the inner cylinder.

My attempt:
To find the electri field strength E I first tried to find an expression for the charge and integrate it over the limits r and 0, but I'm having problem getting the charge expression. Help would be hugely appreciated.

Jan 12th 2009, 01:27 PM

Kiwi_Dave

Quote:

Originally Posted by free_to_fly

I'm stuck on this question:
A sphere of radius R is charged to a potential V far from any other bodies and is inserted inside the inner conductor of the cylindrical capacitor without touching it.

First consider the sphere with charge Q which starts out physically remote from anything else. We have by Gauss:

Where S is any closed surface enclosing the sphere. The problem has spherical symmetry so if we choose a sphere of radius r for the surface S then the E field is normal to the surface and the same magnitude everywhere. This makes the integral easy to solve:

or

Now the voltage at the point of infinity is zero so we can find the voltage on the surface of the sphere by integrating E from infinity to the surface of the sphere:

or

Now once the sphere is placed inside your capacitor the plates of the capacitor force the E field to be uniform between the concentric plates. Therefore, in this area the field will have cylindrical symmetry. We use Gauss again only this time we choose a cylinder radius r, length l for our integrating surface: